There are fewer results for the irreducibility and ergodicity of solutions of interacting Brownian motions. Albeverio–Kondratiev–Röckner [1]} proved the equivalence of the ergodicity of Dirichlet forms and the extremal property of the associated (grand canonical or canonical) Gibbs measures with potentials of Ruelle's class [2]}.
Corwin and Sun [3]} proved the ergodicity of the Airy line ensembles, for which the dynamics are related to the Airy\( _2\)  random point field.
The first author and Tsuboi [4]} proved that the labeled dynamics of Dyson's model in infinite dimensions are irreducible.
A general result concerning the ergodicity of Dirichlet forms can be found in [5]}.
